Most peer-assisted media streaming systems have applied a design philosophy that uses a "mesh" topology of peers: each peer connects to a small number of neighboring peers, with which it exchanges state information periodically. Requests for media segments are made to these neighboring peers on-demand. Apparently, with such a "gossiping" design principle, peers do not make decisions based on global knowledge of the entire system, sacrificing system-wide efficiency. If a peer connects to more neighbors, the gap between local and global knowledge is mitigated; however, the overhead of communicating with all neighbors periodically is proportionally higher. In this paper, using theoretical analysis based on a system of difference equations, we show the surprising result that, once the number of neighbors exceeds a very small threshold (typically 5), the peer upload capacity is fully utilized. We demonstrate that such a threshold is not affected by the scale of system, a...